I assume that the command travels from my code to JOGL, and then to the ATI graphics driver which then breaks this OpenGL "command" down into pieces that the graphics card can understand. Is this correct?
Yup, the driver translates it into hardware-specific code. The Catalyst driver (for ATI) can generate different code for a 9600, 9700 or a X800 express, it's up to the driver. So inside your driver-installer, you will find all drivers for all series of cards, that's one of the reasons it's so big. That hardware specific code is wrapped in a protocol and sent to the gfx-card.
About the slowdowns you experience: try to install the latest drivers, but Vista translates all OpenGL calls into DirectX, in windowed-mode. If you see major speedups in fullscreen mode (if there is such a thing in that app), it might be the wrapper-overhead.